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In the derivation of the Black-Scholes equation, I see that a portfolio $\Pi=V-\Delta S$ is used, where $\Delta$ turns out to be $\frac{\partial{V}}{\partial S}$. But to determine $\Delta$, it is assumed to be constant. So my confusion comes from the reasoning why $\Delta$ is constant as when I look at the graphed solutions for a call option, $\frac{\partial{C}}{\partial S}$ is clearly not constant when S is less than the price of the strike price.

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  • $\begingroup$ Is your $y$-axis $C$, $V$ or $\Pi$? It may be worth editing in a proof that $\partial_S V$ being constant would imply $\partial_S C$ is too. $\endgroup$ – J.G. Dec 27 '18 at 17:51
  • $\begingroup$ @J.G. the y-axis is C- the call option value and in derivations, people state $d\Pi=dV-\Delta dS$ implying |delta is a constant. $\endgroup$ – DLB Dec 27 '18 at 19:37

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